Background
One may study the convergence of series whose terms an are elements of an arbitrary abelian topological group. The notion of absolute convergence requires more structure, namely a norm, which is a real-valued function on abelian group G (written additively, with identity element 0) such that:
- The norm of the identity element of G is zero:
- For every x in G, implies
- For every x in G,
- For every x, y in G,
In this case, the function induces on G the structure of a metric space (a type of topology). We can therefore consider G-valued series and define such a series to be absolutely convergent if
In particular, these statements apply using the norm |x| (absolute value) in the space of real numbers or complex numbers.
Read more about this topic: Absolute Convergence
Famous quotes containing the word background:
“... every experience in life enriches one’s background and should teach valuable lessons.”
—Mary Barnett Gilson (1877–?)
“Silence is the universal refuge, the sequel to all dull discourses and all foolish acts, a balm to our every chagrin, as welcome after satiety as after disappointment; that background which the painter may not daub, be he master or bungler, and which, however awkward a figure we may have made in the foreground, remains ever our inviolable asylum, where no indignity can assail, no personality can disturb us.”
—Henry David Thoreau (1817–1862)
“They were more than hostile. In the first place, I was a south Georgian and I was looked upon as a fiscal conservative, and the Atlanta newspapers quite erroneously, because they didn’t know anything about me or my background here in Plains, decided that I was also a racial conservative.”
—Jimmy Carter (James Earl Carter, Jr.)